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Gibbs-Duhem Relation/Entropic Equations of State

  1. Nov 1, 2013 #1
    I'm working through Callen (ch. 3.4) right now and I'm trying to follow his explanation of the Gibbs-Duhem relation:

    Beginning with the ideal gas characterizations
    [tex]
    PV = nRT
    [/tex]
    [tex]
    U = cnRT
    [/tex]
    there is some rearranging of terms and we are shown the relations
    [tex]
    \frac{P}{T}=\frac{nR}{V}
    [/tex]
    [tex]
    \frac{1}{T}=\frac{cNR}{U}
    [/tex]
    and recognizing the further definitions of [itex]n/V \equiv v[/itex] and [itex]N/U \equiv u [/itex].
    Then Callen states 'from these two entropic equations of state we find the third equation of state'
    [tex]
    \frac{\mu}{T} = \,function\, of\, u,\, v \,\,\,\,\,\,[1]
    [/tex]
    by integrating the Gibbs-Duhem relation
    [tex]
    d\left(\frac{\mu}{T}\right) = ud\left(\frac{1}{T}\right) + vd\left(\frac{P}{T}\right)
    [/tex]

    I'm not seeing where he gets [1] from. Is that just a definition? I noticed a few sections before this he defined:
    [tex]
    d\mu = -sdT + vdP
    [/tex]
    I'm missing something but I don't know what. Just looking for some clarification, thanks.
     
    Last edited: Nov 1, 2013
  2. jcsd
  3. Nov 2, 2013 #2

    dextercioby

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    Science Advisor
    Homework Helper

    He says that the integrated version [1] comes from the differential one called Gibbs-Duhem. So your question is actually where Gibbs-Duhem comes from and that should be answered by Callen.
     
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